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Steve_BradyFlag for United States of America

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Mathematical relationship between two line segments perpendicular to a circle

Hello,

What is the mathematical relationship between the length of two line segments, each of which extends radially (perpendicularly) from a separate point on a circle to a line which is tangent to a third point on the circle?

For example, the following diagram shows a circle with origin o and radius r (not = 1) which is centered on the intersection of horizontal axis x and vertical axis y. Line z is tangent to the circle at point p which is the top intersection of the circle and vertical axis y. Line segments s & t (red & green respectively) each begin on the circle and extend radially to line x. Angle a is formed by the vertical axis y and the radial extension of line segment s while angle b is formed by the radial extensions of line segments s & t. Arc d extends from the origin of line segment s to the origin of line segment t.


Questions:

1) What is the mathematical relationship between the length of line segments s & t?

2) What is the length of arc d (may be expressed in radians)?

Thanks
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ozo
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If it helps you to think about this from first principles, you have two right angle triangles Y, o, s and Y, o, t with a third triangle embedded s, t and a segment of the tangent. Since two of the triangles are right angled, you can work out all the numbers from first principles if you wish. It comes out to #1 in the above post.

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Thanks for the response.

Could you show the derivation please?

I understand that

        secant = hypotenuse/adjacent

but I have not been able to determine how you arrived at that solution.
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Many thanks.